We consider FK percolation on \mathbb{Z}^d with interactions of infinite range of the form J_x = \psi(x)\mathsf{e}^{-\rho(x)} with \rho a norm on \mathbb{Z}^d and \psi a subexponential correction. We first provide an optimal criterion ensuring the existence of a nontrivial saturation regime (that is, the existence of \beta_{\rm sat}(s)>0 such that the inverse correlation length in the direction s is constant on [0,\beta_{\rmsat}(s))), thus removing a regularity assumption used in a previous work of ours. Then, under suitable assumptions, we derive sharp asymptotics (which are not of Ornstein-Zernike form) for the two-point function in the whole saturation regime (0,\beta_{\rm sat}(s)). We also obtain a number of additional results for this class of models, including sharpness of the phase transition, mixing above the critical temperature and the strict monotonicity of the inverse correlation length in \beta in the regime (\beta_{\rm sat}(s), \beta_{\rm c}).
